System identification means estimating a mathematical model (transfer function, impulse response, etc.) of an input/output relation of a system based on input/output data, and typical application examples include an echo canceller in international communication, an automatic equalizer in data communication, an echo canceller and sound field reproduction in a sound system, active noise control in a vehicle etc. and the like. Hitherto, as an adaptable algorithm in the system identification, LMS (Least Mean Square), RLS (Recursive Least Square), or Kalman filter is widely used. In general, an observed value of output of a system is expressed as follows:
                    [                  Mathematical          ⁢                                          ⁢          Expression          ⁢                                          ⁢          1                ]                                                                      y          k                =                                            ∑                              i                =                0                                            N                -                1                                      ⁢                                          h                i                            ⁢                              u                                  k                  -                  i                                                              +                      v            k                                              (        1        )            Where, uk denotes an input, hi denotes an impulse response of the system, and vk is assumed to be a white noise.
The details are described in non-patent document 1 or the like.
1. LMS
In the LMS, an impulse response xk=[h0, . . . , hN−1]T of a system is estimated from an input uk and an output yk as follows:[Mathematical Expression 2]{circumflex over (x)}k={circumflex over (x)}k−1+μHkT(yk−Hk{circumflex over (x)}k−1)  (2)Where, Hk=[uk, . . . , uk−N+1]T, μ>0.2. RLS
In the RLS, an impulse response xk=[h0, . . . , hN−1]T of a system is estimated from an input uk and an output yk as follows:
                    [                  Mathematical          ⁢                                          ⁢          Expression          ⁢                                          ⁢          3                ]                                                                                  x            ^                    k                =                                            x              ^                                      k              -              1                                +                                    K              k                        ⁡                          (                                                y                  k                                -                                                      H                    k                                    ⁢                                                            x                      ^                                                              k                      -                      1                                                                                  )                                                          (        3        )                                          K          k                =                                            P                              k                -                1                                      ⁢                          H              k              T                                            ρ            +                                          H                k                            ⁢                              P                                  k                  -                  1                                            ⁢                              H                k                T                                                                        (        4        )                                          P          k                =                              (                                          P                                  k                  -                  1                                            -                                                K                  k                                ⁢                                  H                  k                                ⁢                                  P                                      k                    -                    1                                                                        )                    /          ρ                                    (        5        )            Where, x^0−0, P0=ε0I, ε0>0, 0 denotes a zero vector, I denotes a unit matrix, Kh denotes a filter gain, and ρ denotes a forgetting factor. (Incidentally, “^”, “v” means an estimated value and should be placed directly above a character as represented by the mathematical expressions. However, it is placed at the upper right of the character for input convenience. The same applies hereinafter.)3. Kalman Filter
A minimum variance estimate x^k|k of a state xk of a linear system expressed in a state space model as indicated by
                    [                  Mathematical          ⁢                                          ⁢          Expression          ⁢                                          ⁢          4                ]                                                                                  x                          k              +              1                                =                                    ρ                              -                                  1                  2                                                      ⁢                          x              k                                      ,                              y            k                    =                                                    H                k                            ⁢                              x                k                                      +                          v              k                                                          (        6        )            is obtained by a following Kalman filter.
                    [                  Mathematical          ⁢                                          ⁢          Expression          ⁢                                          ⁢          5                ]                                                                                                x              ^                                      k              |              k                                =                                                    x                ^                                            k                |                                  k                  -                  1                                                      +                                          K                k                            ⁡                              (                                                      y                    k                                    -                                                            H                      k                                        ⁢                                                                  x                        ^                                                                    k                        |                                                  k                          -                          1                                                                                                                    )                                                    ⁢                                  ⁢                                            x              ^                                                      k                +                1                            |              k                                =                                    ρ                              -                                  1                  2                                                      ⁢                                          x                ^                                            k                |                k                                                                        (        7        )                                                      K            k                    =                                                    Σ                ^                                            k                |                                  k                  -                  1                                                      ⁢                                                            H                  k                  T                                ⁡                                  (                                      ρ                    +                                                                  H                        k                                            ⁢                                                                        Σ                          ^                                                                          k                          |                                                      k                            -                            1                                                                                              ⁢                                              H                        k                        T                                                                              )                                                            -                1                                                    ⁢                                  ⁢                                            Σ              ^                                      k              |              k                                =                                                    Σ                ^                                            k                |                                  k                  -                  1                                                      -                                          K                k                            ⁢                              H                k                            ⁢                                                Σ                  ^                                                  k                  |                                      k                    -                    1                                                                                                          (        8        )                                                      Σ            ^                                              k              +              1                        |            k                          =                                            Σ              ^                                      k              |              k                                /          ρ                                    (        9        )            Where,[Mathematical Expression 6]{circumflex over (x)}1|0=0, {circumflex over (Σ)}1|0=ε0I, ε0>0  (10)xk: a state vector or simply a state; unknown, and this is an object of estimation.yk: an observation signal; an input of the filter and known.Hk: an observation matrix; known.vk: an observation noise; unknown.ρ: a forgetting factor; generally determined by trial and error.Kk: a filter gain; obtained from a matrix Σ^k|k−1.Σ^k|k: corresponding to a covariance matrix of an error of x^k|k; obtained by a Riccati equation.Σ^k+1|k: corresponding to a covariance matrix of an error ofx^k+1|k; obtained by a Riccati equation.Σ^1|0: corresponding to a covariance matrix of an initial state; although unknown, ε0I is used for convenience.
In addition, hitherto, there are techniques described in patent documents 1 and 2 and non-patent documents 2 to 5.    Patent document 1: WO 02/035727, JP-A-2002-135171    Patent document 2: WO 2005/015737    Non-patent document 1: S. Haykin, Adaptive Filter Theory, 3rd Edition, Prentice-Hall, 1996    Non-patent document 2: K. Nishiyama, Derivation of a Fast Algorithm of Modified H∞ Filters, Proceedings of IEEE International Conference on Industrial Electronics, Control and Instrumentation, RBC-II, pp. 462-467, 2000    Non-patent document 3: K. Nishiyama, An H∞ Optimization and Its Algorithm for Time-Variant System Identification, IEEE Transactions on Signal Processing, 52, 5, pp. 1335-1342, 2004    Non-patent document 4: B. Hassibi, A. H. Sayed, and T. Kailath, Indefinite-Quadratic Estimation and Control, 1st Editions, SIAM, 1999    Non-patent document 5: G. Glentis, K. Berberidis, and S. Theodoridis, Efficient least squares adaptive algorithms for FIR transversal filtering, IEEE Signal Processing Magazine, 16, 4, pp. 13-41, 1999